Potential energy function

when a particle moves in an attractive central force field with a potential enery function V(x) = -k/r, for a gives non-zero value of angular momentum, at a certain time there is a minimum energy for which,it is possible to find the solutions to the equations of motions. at this minimum energy the particle moves in an elliptical orbit.

It's not the most lucid description I've ever seen, but what I think it means to say is that, for a particle in an inverse-square force field (I`ll take a star and a planet), the trajectory of the planet is an ellipse under a few assumptions:
- The planet does not head straight towards the sun (non-zero angular momentum)
- The total energy of the particle must be negative.

If the total energy is positive it will not get caught into an orbit but can reach infinity (where V=0) with kinetic energy left. The trajectory will be a hyperbola.
If the total energy is (exactly) zero you have a special case and the trajectory is a parabola.